That is, as you suggest, a feature of IE – not a general feature of Windows. IE is doing some malarky under the hood – receiving an authentication prompt from the web server, and then responding to it with a security token. You'd have to build that functionality yourself, and it ain't gonna be easy. Invoke-WebRequest is a much lower-level creature, and doesn't do any of the fancy stuff a browser does – like track cookies, automatically respond to auth requests, etc. It's just sending data back and forth.
Instead of -Credential, you'd just use the -UseDefaultCredentials switch on Invoke-WebRequest. This is assuming that your website uses Windows Authentication. If it's HTTP auth or forms-based or whatever, then you can't use the user's existing credentials without them re-entering the password.
@dlwyatt – Thank you very much. Was exactly what I was looking for. Don't know how I missed the -UseDefaultCredentials parameter. I am using Windows Authentication – sorry for not making that clear in the first place.